Pop stars, pop artists - that is where it all begins, the romance between contemporary music and art. It is no coincidence that two of the defining images of the 60s just happen to be love letters to the greatest voices (and faces) of the day: Mick Jagger and Elvis Presley.

Andy Warhol's Elvis (1963), with his ominous stare, is literally printed on a silver screen - a still of a movie image. In one half of the diptych he performs in sharp black and white, in the other he is already dissolving into the glittering shadows of legend. And Richard Hamilton's deathless Swingeing London 67, showing Jagger cuffed in the back of a police van after being sentenced for possession - the handcuff solid sparkling silver - is just as much of a votive icon. The establishment's last attempts to hold back the tide, the beautiful rebel shielding his eyes against the paparazzi glare: songs of revolution in the air.

From the 1950s onwards, our art colleges may have nurtured some of the most influential names in music. But in the past couple of decades, the creativity has gone both ways to such an extent that there is now time-honoured continuum.

There are artists who have a parallel career as performers. Sam Taylor-Wood recently released a breathy cover version of I'm in Love with a German Film Star produced by Pet Shop Boys, for whom she has directed videos and concert films. Jim Lambie, shortlisted for the Turner Prize in 2005 for his wild floor installations made of multicoloured electrical tape, was in Glasgow band the Boy Hairdressers, two members of which went on to start Teenage Fanclub. He is also a DJ, like the Beck's Futures' winner Matt Stokes and even the Chapman brothers, who once curated an All Tomorrow's Parties festival at Pontin's Camber Sands featuring Aphex Twin, Lightning Bolt and Throbbing Gristle.

The Turner Prize winner Martin Creed plays in a band called Owada, who released their first delicate and eccentric album, Nothing, on Piano, the label started by David Cunningham, conceptual artist and the man behind the Flying Lizards. The artist Edwin Pouncey - aka Savage Pencil - has designed sleeves for the Fall, Sonic Youth and Big Black and made his own records with drone-rock group Pestrepeller.

Obviously, artists who can't play can make something else of their musical obsessions. Jeremy Deller, currently working with Bolton's Blackout Crew, based an entire show on the Manic Street Preachers, the members recreated in Lego, with a paean commissioned in their honour by the Eisteddfod Bard. A Los Angeles-based artist, Julie Becker, spent months in an editing suite trying to prove the urban myth that Dark Side of the Moon was an alternative soundtrack to The Wizard of Oz by synching the two (insufficient evidence, in the end). Another Turner winner, Douglas Gordon, made a vast installation from bootleg videos of concerts by the Smiths, the Cramps and the Rolling Stones. Slowed, manipulated, shifting past at a few frames per second, they appear to stop time: the fan's dream in the mosh pit.

Gavin Turk's life-size statue of himself in the role of Sid Vicious, which in turn reprises the pose of Elvis in Warhol's diptych, is both a remix and a wishful fancy. Music inspires a fantastical strain in contemporary art. Think of Dexter Dalwood's paintings of the interiors of pop stars' homes, including Paisley Park and Neverland (Michael Jackson's Bedroom), places he has never been but imagines in florid, if deserted, detail; the pictures play on pop myths as well as the music itself.

Elizabeth Peyton's paintings of stars - Eminem, Kurt Cobain, Jarvis Cocker - try to get beyond celebrity photos. She imagines John Lennon as a winsome little boy, Liam and Noel in their Sunday best on mum's sofa, Jarvis having a polite cigarette. Perhaps the best painted portrait in recent years is Gary Hume's Michael (2001), simplified, laconic and done in a gloss paint that perfectly evokes the vacuous quality of Michael Jackson's increasingly shocking mask of a face.

Art's relationship with music tends to go two ways. Either it is overt homage to particular performers - in the case of the Turner Prize-winning painter Chris Ofili, roll-calls of black icons such as James Brown, Louis Armstrong and Little Richard, name-checked in his trademark clumps of elephant dung - or it is a fascination with the whole culture. The best-loved work of Mark Leckey, last year's Turner Prize winner (are you spotting a trend?), is probably Fiorucci Made Me Hardcore (1999), a dream-like montage of dancing crowds at various clubs that covers a period of 30 years, from northern soul to hardcore raves. The best of these quasi-anthropological films is Matt Stokes's Long After Tonight (2006), a video of a northern soul night in a Dundee church, beautifully choreographed and filmed, comparing two kinds of ritual behaviour. The camera is at knee height, then hip height, watching the dancers drop, spin and swirl. It starts with a hemline twirling in slow motion, ends with dawn, and conjures flamenco, waltz, disco, ballet and dervish-whirling - and everything in between.

There is so much more music-minded art around - Tracey Emin and Gillian Wearing as disco divas, the MTV videos of Chris Cunningham now collected by art museums, Luke Fowler's fake pop documentaries, Matthew Barney's disastrous casting of his wife Björk in the Cremaster Cycle, the magnificently eerie installations of those two sound-and-light magicians Janet Cardiff and George Bures Miller - but one work completely sums up the balance of the relationship, art more in love with music than vice versa. This is Jamie Shovlin's terrifically artful archive of Lustfaust memorabilia from 2006 - Lustfaust being a post-punk cult band from Germany. You got their music by sending them a blank cassette (they despised the industry) and designing your own label. They never recorded an album. And Shovlin could only play 30-second extracts of their bootlegs - imagine thrash Lou Reed - because of threats from acrimonious ex-members.

Lustfaust Biroed on a pair of plimsolls, Melody Maker ads, reviews of underground gigs, excitable postcards sent back and forth between fans - a whole era was evoked in the archive, as well as memories of one's own age of obsession. (You can still see it at lustfaust.com.) How wonderful if Shovlin had made it all up, you thought, and it turned out he had.